Discrete Probability Distributions

Main Points

• Overview of Discrete Random Variables and Probability Distributions
• Example: Binomial Distribution
• Example: Poisson Distribution

Contents

Overview of Discrete Random Variables and Probability Distributions

• Discrete random variables can take on integer values only and can be represented in tables.

  Given $S=\{x_1,x_2,...,x_n\}$, where $S$ is the set of all domain values, $X$ is the discrete random variable denoting a value in set $S$, and $n$ is the number of these values in set $S$,

  Denote $P$ as the set of all probabilities corresponding to each domain value, where

  $P=p_i=P(X=x_i),i\in[1, n]$,

Denote E(X) as the expected value of X

E(X) represents the discrete probability distribution's central tendency (mean).

Denote Med(X) as the median value of X, given S is an ascendingly ordered set,

• $Med(X)$ represents the discrete probability distribution’s middle value where

  $\dfrac{1}{2}=P(X\le Med(X))=P(X\ge Med(X))$.

  Denote $\sigma^2$=Var(X) as the variance of X​

• $\sigma^2$ represents the discrete probability distribution’s spread.

Denote $\sigma=SD(X)$ as the standard deviation of $X$,

$\sigma$ represents the discrete probability distribution’s standardised spread relative to the mean.

Binomial Distribution

• Properties

  • A binomial experiment consists of $n$ repeated and independent trials.
  • The result of each trial can have two outcomes only.
  • The probability of success, $p$, and the probability of failure, $q$, are constant in each trial.

The expected value, variance and standard deviation of the binomial distribution are as follows.

Given$X\sim Bin(n,p)$, denote P(X=x) as the probability of having exact x successes from n trials,

• The Bernoulli distribution is a subset of the binomial distribution where $n=1$.

  Denote $P(X=1)$ as the probability of having exact $1$ success from $1$ trial,

Denote P(X=0) as the probability of having exact 0 success from 1 trial,

Poisson Distribution

• Properties

  • A Poisson distribution is a variant of the binomial distribution with no upper limit to the number of trials.

    • $\lambda$ is used to denote the mean of the Poisson distribution.

      denote $P(X=x)$ as the probability of observing $x$ events in the given interval,

    • It concerns the probability of an occurrence which occurs in a specific interval.

    • Occurrences are independent of each other and they do not overlap.

  • A Poisson process is a stochastic process in which events occur continuously and independently, each having a small probability of occurrence, $p$.

    • The interval of values can be very small (which could approximate infinitesimal amounts), where each specific value has a small probability of occurrence.

  • The expected value, variance and standard deviation of the Poisson distribution are as follows.

Given $X\sim Pois(\lambda), let X(t_a,t_b)$ be the number of events per interval [t_a,t_b],